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\title{An Instrument-Pricing on A Specific Unbalanced Cooperative Games: Machine Scheduling Game}
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The IPU game is very special so that we can use SPT rule to compute the $c_{IPU}(s)$ , which means jobs in coalition $s$ with the shortest processing time should be processed first and the longest ones last. Then we can use dynamic programming to solve the separation problem $\delta_{IPU} = \min \big\{c_{IPU}(s)+P-\bar{\alpha}(s,P): \forall s \in \mathbb{S} \setminus \{N\}\big\} $

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